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We introduce the notion of a k-synchronized sequence, where k is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be k-synchronized if its graph is represented, in base k, by a right synchronized rational relation. This is an intermediate notion between k-automatic and k-regular sequences. Indeed, we show that the class of k-automatic sequences is equal to the class of bounded k-synchronized sequences and that the class of k-synchronized sequences is strictly contained in that of k-regular sequences. Moreover, we show that equality of factors in a k-synchronized sequence is represented, in base k, by a right synchronized rational relation. This result allows us to prove that the separator sequence of a k-synchronized sequence is a k-synchronized sequence, too. This generalizes a previous result of Garel, concerning k-regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.

@article{Carpi2001, abstract = {We introduce the notion of a $k$-synchronized sequence, where $k$ is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be $k$-synchronized if its graph is represented, in base $k$, by a right synchronized rational relation. This is an intermediate notion between $k$-automatic and $k$-regular sequences. Indeed, we show that the class of $k$-automatic sequences is equal to the class of bounded $k$-synchronized sequences and that the class of $k$-synchronized sequences is strictly contained in that of $k$-regular sequences. Moreover, we show that equality of factors in a $k$-synchronized sequence is represented, in base $k$, by a right synchronized rational relation. This result allows us to prove that the separator sequence of a $k$-synchronized sequence is a $k$-synchronized sequence, too. This generalizes a previous result of Garel, concerning $k$-regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.}, author = {Carpi, Arturo, Maggi, Cristiano}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications}, keywords = {regular sequence; automatic sequence; separator; -synchronized sequence}, language = {eng}, number = {6}, pages = {513-524}, publisher = {EDP-Sciences}, title = {On synchronized sequences and their separators}, url = {http://eudml.org/doc/92681}, volume = {35}, year = {2001},}

TY - JOURAU - Carpi, ArturoAU - Maggi, CristianoTI - On synchronized sequences and their separatorsJO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et ApplicationsPY - 2001PB - EDP-SciencesVL - 35IS - 6SP - 513EP - 524AB - We introduce the notion of a $k$-synchronized sequence, where $k$ is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be $k$-synchronized if its graph is represented, in base $k$, by a right synchronized rational relation. This is an intermediate notion between $k$-automatic and $k$-regular sequences. Indeed, we show that the class of $k$-automatic sequences is equal to the class of bounded $k$-synchronized sequences and that the class of $k$-synchronized sequences is strictly contained in that of $k$-regular sequences. Moreover, we show that equality of factors in a $k$-synchronized sequence is represented, in base $k$, by a right synchronized rational relation. This result allows us to prove that the separator sequence of a $k$-synchronized sequence is a $k$-synchronized sequence, too. This generalizes a previous result of Garel, concerning $k$-regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.LA - engKW - regular sequence; automatic sequence; separator; -synchronized sequenceUR - http://eudml.org/doc/92681ER -